In this post, Kevin O’Bryant has considered “least nonzero digit”-type examples . Encouraged by his remark that there are only very few possibilities, i.e. feasible sets S, for each modulus m I did a complete enumeration up to modulus 200. The goal was (and still is) to find a slowly growing example possibly beating the current record holder (please read my last post for more information).

Much to my surprise there are certain “magic” numbers with comparatively many feasible sets S. What follows is a table (modulus m: number of sets S) for those m with more than 4 sets:

For example, there is no set S for m=105. This can be seen as follows: Logarithmic discrepancy of implies for . Since complete multiplicativity implies . Now and again since is completely multiplicative produces a contradiction.

What makes these number “special”? Why is there this huge jump from 4 to “many”? Is this just one more “random” number theoretic artifact or can we understand this by considering how the m-examples are constructed from the d|m-examples?

I do not know the answers, but let me close with something I do know. In my last post I gave a proof that any that grows slower than must have . Subsequent computer searches have increased that bound to 1000.

In this post I have collected results of the fifth Polymath project on the Erdös discrepancy problem. With hundreds and maybe even thousands of posts I have decided to restrict this to the (maybe) simplest non-trivial special case, namely completely multiplicative functions with values in . That has some justification since there are results reducing the problem to the completely multiplicative case (albeit with values in ). Furthermore my focus is on elementary examples related to the problem and how far one can get with textbook methods. It that sense this can be seen as a beginner’s guide with the quoted results being nothing more than exercises. I have arranged the results within the following sections

Statement of the problem

Examples related to the problem

Necessary conditions for bounded discrepancy

Discrepancy results

What fascinates me about the problem is that it can be stated without much technicalities and that a solution seems to be within reach due to recent results in number theory.

Discrepancy Problem. Find or prove the non-existence of a function with the following properties:

is completely multiplicative, i.e. for all .

has bounded partial sums (bounded discrepancy), i.e. there is a such that for all .

Let me fix some notation before I present examples related to the above question. In the following denotes a completely multiplicative function with values in , denote primes and is a complex number with real part and imaginary part .

Dirichlet characters of modulus are completely multiplicative with values in . Since Dirichlet characters assume the value they are strictly speaking not in the focus of this post. However, they can be used to construct -valued examples (cf. the next example) and they play a central role in the discussions.

The functions and with being the quadratic Dirichlet character mod (Legendre symbol) and an odd prime.

To generalize the last example fix and a set . Define if the last non-zero digit of in its -ary representation is in . In all other cases . Immediately from the definition we find if . To not obtain unbounded discrepancy it is necessary that for . Moreover, we find which implies logarithmic growth. Since is completely multiplicative we have if n is a quadratic residue mod , implying for an odd prime. We construct some for small composite :

For we have for the quadratic residues and thus depending on whether we include either or .

For we get . Since for all we cannot include and . Because is included this forces for all . Now using complete multiplicativity and we get the four cases: , , and .

For we have that . Thus . Since we have , a contradiction.

Necessary conditions for bounded discrepancy.

If has bounded partial sums, then the Dirichlet series converges for . Proof idea: Integration by parts.

If has bounded partial sums, then . Proof idea: One first establishes like in the proof of Dirichlet’s theorem on primes in arithmetic progressions. This estimate is then used in the textbook proof on the growth of .

If has bounded partial sums, then for infinitely many primes one has and for infinitely many primes one has . Proof idea: Otherwise one would get a contradiction to the standard result on the growth of .

With not so elementary methods one can prove results ‘in the spirit of’: Let have bounded partial sums and let be a non-principal Dirichlet character. Then . Here more serious number theory kicks in and this is probably where most are currently heading to.

Discrepancy results.

Computer experiments (no link found) show that there are with for all . No such sequence exists for . There is no traditional proof of this, not even for .

Liouville’s function does not have bounded partial sums since its Dirichlet series equals and thus has a singularity at .Alternatively one can proceed as in the proof of Dirichlet’s theorem on primes in arithmetic progressions and apply Dirichlet’s hyperbola method to ultimately showing that under the assumption of bounded partial sums grows like . This is a contradiction since grows logarithmically. Showing that the discrepancy of the Liouville function is for all is equivalent to solving the Riemann hypothesis.

For primitive Dirichlet characters mod one has . Proof idea: Apply partial summation to the associated Gauss sum and use .

We have and with equality holding only for and .

is optimal in the following sense: with equality holding only for . Proof idea: One constructs a sequence such that with and . With the lower bound for the discrepancy of primitive Dirichlet characters one can eliminate all but finitely many cases. These can then be checked by computer. A similar argument shows that the sums of grow faster than .

Computer experiments so far did not find a that grows slower than . For a given let . Then with is the largest such that there is an with . Proof idea: Do a computer search among all completely multiplicative discrepancy 2 functions with domain ( must be less than 247). A beating thus has at least and thus implying .

Last year I have spent some time to think about the Erdös Discrepancy Problem (EDP). My interest was sparked by Tim Gower’s Polymath project and two comments of Terence Tao (1, 2) on the special case of completely multiplicative functions. Let me state this version of EDP.

Problem. Prove or disprove the existence of a function with the following properties

is completely multiplicative, i.e. for all .

has bounded partial sums, i.e. there is a such that for all .

In his comments Terence Tao describes how one can use ideas of Dirichlet’s proof (of the theorem on primes in arithmetic progressions) to get first an elementary proof of the unboundedness of the summatory Liouville function and second a necessary condition for unbounded discrepancy using properties of . While the ‘positivity’ part is straight forward I had some trouble to fully digest the complex analysis part at the end of his second comment (bounded discrepancy implies ). However, all this can be achieved with textbook arguments and I took this as an opportunity to learn some (analytic) number theory. In what follows I elaborate some ideas on the analytic number theory approach and (in an upcoming post) on the positivity approach (Dirichlet’s idea).

Analytic Method. To show where we can apply complex analysis let me first repeat a standard argument. Define with being the von Mangoldt function. Plugging in the definition yields

Separating primes with from primes with we find (using )

Setting for all we find that and since we find for all

Defining and we arrive at the

Proposition.

The above argument indicates how to proceed. Using analytic number theory similar to the proof of the prime number theorem (PNT) one gets information on for general . It might even be possible to prove that having bounded partial sums implies , although I have not yet checked the details. If true, we would get a nice analog to the PNT in the (still maybe empty) case of bounded partial sums.

Warning to teaching staff: summing infinite sequences of positive (>0) integers is difficult, but possible. For example is not a mistake and students must get full marks. Let me show you why.

A couple of months ago I was hoping to ‘cheat’ a proof of the Erdös discrepancy conjecture by using a variant of an idea of Fürstenberg’s proof on the infinitude of primes. Remember, Fürstenberg considered the integers with a new topology. Its open sets are for and .

This topology is metrizable. There are a couple of hand-waving arguments how this metric could look like. However, as far as I am aware of, there is so far no neat description in the literature. A couple of days ago, R. Lovas and I. Mezö have published a fairly straightforward proof that with induces the above topology.

Since the sequence converges to 0 in this topology. The partial sums of satisfy and thus .

R. Lovas and I. Mezö have collected more such observations in their note. What they did not mention explicitly, but what I consider interesting is that with the above metric, the integers become an ultrametric space. Without loss of generality we assume . Then are all divisors of m and n and thus they are divisors of m+n. Therefore . The strong triangle inequality now follows .

Its Polymath time. My resources, especially time and most importantly skill, are limited and therefore I have to restrict myself a little. Let me just sketch where I set the boundaries, what I want to try and what a possible (successful) outcome might look like. Some acquaintance with Tim Gowers’s proof of Roth’s theorem and its (hoped for) connection to EDP is necessary.

What is the idea?

Let me first collect some observations:

‘Translation’ acts as a group on APs and is periodic on APs with common difference.

ROI starts with some representation of the translation group in terms of rank one projections using exponentials.

An elementary formula from Fourier Analysis describing the interaction of translation and exponentials is used to express the exponentials in terms of Fourier coefficients of some characteristic function and its translates.

The result is an ‘efficient’ representation, i.e. it allows to deduce unbounded discrepancy.

Translation as described above lives on the domain of the Fourier coefficients. We do not lose information if we consider it for the corresponding Fourier series. On all reasonable spaces translations form a strongly continuous (semi-) group of linear operators. For periodic strongly continuous groups we have representations of the group. If we work with rotation groups on with we even have a tensorial representation in terms of exponentials. By the way, if we choose our space carefully, exponentials are at least approximative eigenvectors of translations (rotations).

The idea now is to get information on discrepancy on (the domain of the Fourier coefficients) by studying ROI on various spaces of Fourier series, like or .

If I am not able to translate the ROI idea to the infinite section directly I will also try to use the so called ‘Complex Inversion Formula’, expressing the integral over the group in terms of some inverse Laplace transform. This can be seen as an infinite dimensional version of Perron’s formula. However that would be a ‘last try’ since it is not connected to Polymath anymore.

What is the goal?

The result I am aiming for looks roughly as follows:

Let be a Banach (or better a Hilbert) space and let be a strongly continuous (maybe only semi-)group of linear operators (translations, rotations, periodic?) with generator . If the resolvent satisfies some conditions (including norm estimates) then ‘something’ has unbounded ‘discrepancy’.

That sounds incredibly naive, since Tim Gowers’s proof needed some clever estimates and preyed upon cancellations for different common differences . My hope is to hide these technicalities in an estimate on the resolvent. Such estimates like e.g. in the Hille-Yosida theorem are usually harder to obtain than to state. That is good news.

So far, that would be the first part. In a way this is just translating the main idea of the proof into some other language. The hard part would then be to apply the general theorem to other situations (maybe even EDP). Here the resolvent has to be estimated and technicalities enter. However, I do not want to think that far ahead.

So far that was a hot summer. With all written and oral exams finished and with bearable temperatures returning I plan to spend some free time to think about the Erdös Discrepancy Conjecture. Is it wise to post this plan? I think so. It puts some pressure on me and this is good.

The idea is that we have probably not made maximal use of Tim Gowers’s proof of Roth’s theorem. Behind all the numbers and estimates there could be some abstract/infinitary content. If this is the case, then it might be easier to generalize or at least to spot the limits of the method.

Intuition or wishful thinking? Maybe the later, but I want to know for sure.

Two weeks of constant snowing is highly unusual for Germany. It makes driving a real adventure especially since my home town ran out of salt a week ago.

While being snow-covered in endless traffic jams I was not lazy and added up +1’s and -1’s to get an idea of what is going on here: Polymath5 project. The Erdös discrepancy problem is easy to formulate and really hard to solve. Give it a try!